## Abstract

The enhanced nonlinear interactions that are driven by surface-plasmon resonances have readily been exploited for the purpose of optical frequency conversion in metallic structures. As of yet, however, little attention has been payed to the exact particulate nature of the conversion process. We show evidence that a surface plasmon and photon can annihilate simultaneously to generate a photon having the sum frequency. The signature for this nonlinear interaction is revealed by probing the condition for momentum conservation using a two-beam k-space spectroscopic method that is applied to a gold film in the Kretschmann geometry. The inverse of the observed nonlinear interaction—an exotic form of parametric down-conversion—would act as a source of surface plasmons in the near-field that are quantum correlated with photons in the far-field.

© 2013 Optical Society of America

## 1. Introduction

Surface plasmons (SP) are an oscillation in electronic charge density at the interface between a metal and a dielectric [1]. Their strong confinement to the surface leads to an enhancement of the electromagnetic field. SP have therefore become an interesting approach for sensing techniques [2] and nonlinear optics such as four-wave mixing (4WM) [3] and second-harmonic generation (SHG) [4]. Since the first demonstration of SHG from a smooth metal surface by Simon et al. [5], the phenomenon of harmonic generation involving propagating SP has been elucidated by numerous studies [6–10]. However, relatively few studies have addressed the particulate nature of the nonlinear conversion—whether the plasmon simply enhances the field, or is itself an active partner converted in the interaction. Active participation of SP has been demonstrated in the *χ*^{(3)}-interaction for 4WM on a structured surface [11], and in our previous work on the *χ*^{(2)}-interaction that drives SHG on a smooth metal film [12]. The dominant interaction in the latter was identified as the type where two plasmons at the fundamental create a photon at the second-harmonic. Although a hybrid interaction involving the dual annihilation of a plasmon and a photon was predicted, the single-beam k-space method could not resolve it.

In this paper, we show that the dual annihilation of a plasmon and photon can be driven by degenerate sum-frequency generation (SFG) in a metal film. The conditions for nonlinear phase-matching allowed this interaction to be isolated from others by employing a two-beam k-space spectroscopic method. In contrast to SHG, degenerate SFG is a three-wave mixing process where the three modes of the electromagnetic field do not overlap in k-space. The investigation of the wave-vector projections that fulfill the nonlinear phase-matching conditions was realized using an experimental setup with two incident beams at the fundamental wavelength that had spatial overlap at the metal film, yet were individualy adjustable in terms of their incident angles and femto-second pulse delay. The second-harmonic light emitted by the sample was analyzed in terms of intensity and exit angle, from which the SFG signal and the type of interaction could be extracted. Due to the absence of intricate geometrical effects arising from the nanostructuring of metal films, this versatile method, when applied in the Kretschmann-Raether configuration, is adept at addressing the elementary questions regarding SP propagation and nonlinear interaction.

## 2. Conceptual approach to plasmon-photon wave mixing in k-space spectroscopy

Given that SP generally have a wavevector larger than that of propagating light waves in the surrounding dielectric, SP cannot directly be excited from the far field, though it has been shown that free-space excitation by 4WM is possible [13]. However, there are several options to match the wavevector of the incident light to that of the SP [14]. Due to its simplicity, the Kretschmann geometry is employed here for the excitation, where a collimated light beam interrogates the metal film via a glass prism [15]. Driven as a total internal reflection setup, the larger wavevector in the glass (*ε*_{prism}) reaches the interface between metal (*ε*_{m}) and dielectric (*ε*_{d}) as an evanescent wave.

*θ*

_{in}. When this condition is met, the surface plasmon resonance (SPR) leads to a strong local enhancement of the fundamental field.

To promote SHG from a surface, the nonlinear phase matching conditions have to be taken into account [16]. Similar to SHG in bulk materials [17], the phase of the fundamental field and the related nonlinear polarization have to be matched to the excited second harmonic light’s phase. For the Kretschmann geometry, the maximum intensity of SHG will therefore be obtained for both SP excitation and nonlinear phase-matching.

For the purpose of investigating plasmon-photon interactions in degenerate SFG, two independently adjustable beams are required to ensure conservation of energy and momentum. One can distinguish the nonlinear conversion processes as function of their wavevectors using the following phase-matching conditions

*f*stands for scan-beam photons,

*f′*for auxiliary-beam photons and

*p*for plasmons. The fundamental beam is incident at

*θ*

_{in}while

*θ*

_{out}is the exit angle of the generated second-harmonic radiation. Using Eqs. (2)–(7), the wavevectors of incident and exiting photons, which are functions of the angles

*θ*

_{in}and

*θ*

_{out}, can be related to the corresponding conversion processes as shown in the inset of Fig. 1. As such, each process can be identified by its unique matching combination. In order to conduct an experimental investigation of SFG and SHG with plasmonic participation, Eqs. (3), (4) and (7), that are expected to occur close to the SPR angle, a two-beam k-space spectroscopic method has been applied.

## 3. Experimental setup

The setup shown in Fig. 1 basically consists of two beams, generated by the same laser, that hit a gold-coated prism at individually adjustable angles. Both, the fundamental and harmonic emitted light were detected in k-space and analysed as a function of wavelength and exit angle.

A mode-locked Ti:Sapphire laser tuned to a wavelength of *λ* = 880 nm provided pulses of 150 fs duration at a repetition rate of 76 MHz with a waist radius of 110 *μ*m at the sample. A Mach-Zehnder interferometer split the light into an auxiliary beam *f′* of 20 mW and a scan beam *f* of 60 mW. The delay *τ* was adjustable. The relative phase in the Mach-Zehnder interferometer was dithered over a range of several fringes. The angle of incidence *θ*_{in}, under which the beams impinged on the sample, was varied by two independent travelling mirrors and the beams were focused onto the sample by a 60 mm achromatic lens.

The sample was a BK7-prism with 45nm of 99,99% pure gold evaporated on the hypotenuse side. The outgoing light was collected by another 60 mm achromatic lens while a BG filter removed the majority of fundamental light while transmitting the second-harmonic (at 440nm). A Pellin-Broca prism was used to separate the fundamental and SHG prior to detection on the CCD (5 s integration time for each incident angle, 241 measurements per sample). By using the full 1024×256 pixel CCD-array on which the signal in k-space was projected, we could distinguish the different SHG signals by their location in k-space.

## 4. Results

A map of the second-harmonic intensity produced by the two-beam excitation is shown in Fig. 2 as a function of incident and exit angles. It can be compared to the matching conditions given by Eqs. (2)–(7) to discriminate the different conversion processes.

Figure 2(a) shows the generated second-harmonic light of two beams with a delay *τ* = 1 ps, ensuring that the beams do not interact. The scan beam *f* is scanned from 40° to 50°. The strictly photonic conversion process *ff* − *f*_{2}* _{ω}*, where two incident photons of frequency

*ω*become a 2

*ω*photon, occurs over the whole angle range, thereby forming the diagonal line. When the scan beam approaches the angle for the surface plasmon resonance (

*θ*

_{in}= 42.6°), the harmonic intensity initially decreases and for larger angles

*θ*

_{in}increases two orders of magnitude due to the plasmonic field enhancement. Because of the angular width of the incident beam, plasmons

*p*are generated over a slightly extended range of incident angles and can then participate in nonlinear conversion processes. This leads to the observed horizontal elongation, which is spread over 0.5° of incident angles. It is identified as the

*pp*−

*f*

_{2}

*conversion of two fundamental plasmons*

_{ω}*p*into a harmonic photon [12], which emerges at a constant exit angle

*θ*

_{out}= 42.2°. At angles very near to the diagonal, the

*pf*−

*f*

_{2}

*interaction is also thought to occur (Eq. (3)) and can hence not be discriminated from the*

_{ω}*pp*−

*f*

_{2}

*interaction.*

_{ω}Therefore, only the second-harmonic that considerably deviates from the diagonal trend can precisely be assigned to the *pp* − *f*_{2}* _{ω}* wavevector-matching conditions.

For the investigation of distinct mixing interactions between photons and plasmons, the auxiliary beam *f′* is fixed at *θ*_{in} = 43.7°, which is off-resonant to the SPR and separates both beams sufficiently to avoid overlapping in k-space. The SHG from the auxiliary beam *f′f′* − *f*_{2}* _{ω}* exits at a constant angle of

*θ*

_{out}= 43.2°. There is half a degree offset of the exit angles

*θ*

_{out}for both beams compared with their respective

*θ*

_{in}due to the dispersion in gold and glass.

Figure 2(b) shows the same excitation setup with auxiliary beam *f′* and scan beam *f* overlapping in time (*τ* = 0). The photonic SFG of the two beams *ff′* − *f*_{2}* _{ω}* is visible as an additional diagonal component. This wave-mixing signal shows, as seen in Fig. 2(a) for the scan beam, a horizontal elongation at the scan beam’s SP excitation angle of incidence. This feature is evaluated in Fig. 2(c).

The mixing signal is extracted by subtracting an intensity map of a measurement having a non-zero delay from one taken at zero delay. The elongation and increased intensity of the mixing signal at *θ*_{in} = 42.7° signify the *pf′* − *f*_{2}* _{ω}* process, where an auxiliary beam photon

*f′*and a propagating plasmon

*p*, excited by the scan beam, each of frequency

*ω*become a 2

*ω*photon. As in the former case, the horizontal component is caused by the angular width of both incident beams, thus the scan-beam excites plasmons

*p*over a 0.5° range of incident angles. Since the auxiliary beam

*f′*is held at a fixed angle and the SP’s excited by the scan beam

*f*have a constant wavevector, the generated second-harmonic can only be emitted in one direction. This combination of incident and exit angle and the related wavevectors together with a boost in harmonic intensity is clear evidence of the predicted conversion process.

To confirm our results theoretically, we used a plane-wave Fabry-Pérot model following the analysis of Palomba *et al* [18]. It considers an accumulated nonlinear polarization from both interfaces [19, 20]. The bulk’s contribution to SHG is small compared to that from the surfaces [21] and was therefore not considered in this calculation but it should be taken into account for very high intensities [22]. The nonlinear polarization was calculated at both interfaces from the field components normal to the surface, considering the contributions of two Gaussian beams, that were incident with angles *θ*_{in} and *θ′*_{in}. The dielectric function for gold *ε*_{m} was taken from [23] and the remaining model parameters were derived from the experimental setup, while the relative phase between auxiliary and scan beam was randomly dithered. To ensure correct exit angles *θ*_{out} for the generated harmonic light, matching conditions Eqs. (2)–(7) were applied.

Fig. 2(d) shows the theoretical calculation for two-beam excitation, which is in good agreement with the experimental results. The SFG is clearly visible as additional diagonal component and the SPR leads to strong harmonic intensity at *θ*_{in} =42.3°. Also present is the drop in the SHG-intensity from the scan beam for angles *θ*_{in} just below the surface plasmon resonance. A slight horizontal broadening away from the diagonal in terms of incident angle is visible for scan and SFG beam, whose characteristic strongly depends on the beam waist and therefore the precision of the exciting beam’s wavevector.

## 5. Conclusion

We have performed a k-space spectroscopic measurement of degenerate SFG from a gold film in the Kretschmann geometry. The mixing process between *ω* SP and *ω* photon was clearly isolated by comparing an effective one beam measurement, where both beams were not overlapping in time with a two-beam measurement, that enabled interaction. While the common photonic SHG was measurable over the entire angular range, plasmon-driven SFG was limited to the regime where SP’s are excited and could therefore be clearly distinguished. In particular the *pf′* − *f*_{2}* _{ω}* interaction has been observed. This process could be clearly identified by its unique signature in k-space due to the fact, that only a well-defined combination of wavevectors can take an active part in SFG. The results are in good agreement with our theoretical calculations. Accordingly, we have demonstrated a useful tool that can be applied to investigate particle-quasiparticle nonlinear interactions.

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